# Do people still use Massey Products for computations in the Adams Spectral Sequence

Hey everyone,

It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations.
More "recent" sources (Kahn, Milgram, Ravenel, May (again), Bruner) seem to like to make references to the Massey products, even just for the sake of naming, but use Steenrod Squares for actual computations. The few computations I know of using Massey products can be more easily done with Steenrod Squares, but this may just be my ignorance about Massey products. My question then is do these things still play an important computational role?

Thanks

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But how many people are really doing ASS calculations? If you look at, say, the calculation of the homotopy of tmf at $p=2,3$ then it makes heavy use of Massey products. – Drew Heard Nov 16 '12 at 9:41
Also, if you check out the work of Shimomura and colleagues, who use the chromatic spectral sequence in a fairly heavy way, I'm sure you'll find Massey products a plenty! – Drew Heard Nov 16 '12 at 9:49
Drew, I want to be doing ASS calculations!!!!! But I'm just a little undergrad and there is soooo muuuuuch literature. Thanks for the comment though. – Joseph Victor Nov 16 '12 at 22:14
If you specifically want to see these in action, 'Bordism, Stable Homotopy, and Adams Spectral Sequences' by Kochman, does some serious $p=2$ calculations of the stable homotopy groups of spheres – Drew Heard Nov 17 '12 at 0:00

Why would anything computationally useful be obsolete? Massey products and Toda brackets are intrinsic to stable homotopy theory. It is guaranteed in advance that every element of $E_2$ of the classical Adams spectral sequence (for the homotopy groups of spheres, a similar statement holds more generally for the ASS computing maps between spectra) above the $s=1$ line is decomposable in terms of matric Massey products. Similarly, every element of the stable homotopy groups of spheres is decomposable in terms of matric Toda brackets built from the Hopf invariant one elements. Drew's comments are on the mark: few people nowadays do these kinds of calculations, or know how to do them, which is a pity. We still know relatively little about concrete calculations of stable homotopy groups. These operations are complementary to, not in competition with, Steenrod operations and their related homotopy operations (see e.g. Bruner's contributions to $H_{\infty}$ ring spectra and their applications'').